6.2 Hydrodynamic and particle-particle interface contributions to the effective matrix viscosity
The empirically-found closed form expression used for the effective matrix viscosity for repacking follows Boyer et al. (2011). At constant volume fraction of solids (or alternatively, constant melt fraction), shear and normal stresses scale viscously with macroscopic strain rate (Boyer et al. , 2011). The dependence of shear and normal viscosity on melt fraction is included in eq. (14) and (15). Eq. (14) includes a friction coefficient, \(\mu_{\text{friction}}\), taken to be 0.3 (Boyer et al. , 2011), while both eq. (14) and (15) depend on the maximum packing, \(\phi_{m}\), and are scaled by a reference viscosity, \(\xi_{\text{ref}}\), in eq. (16). The effective matrix viscosity, eq. (16), is the sum of the viscous resistance provided by introducing rigid, non-touching particles into a deforming fluid (hydrodynamic contribution) and that due to touching particles sharing a common interface and dissipating energy by rubbing against one another (friction). Using the most likely solution from the MCMC inversions, we calculate the relative contribution of particle-particle dissipation (includes friction) in the loading curves for the experiments of Hoyos et al. (2022). This quantity is calculated as the relative contribution of eq. (15) and the third term in eq. (14) to the total expression of eq. (16). The relative contribution of particle-particle dissipation for EXP009, for example, is shown inFig. 8 . Fig. 8 demonstrates that the relative contribution of particle-particle dissipation is large compared to hydrodynamic contribution (the relative contribution of the first two terms of eq. (14) to the total expression of eq. (16)). This observation suggests that while at large melt fractions the rheology of magmas may be controlled by hydrodynamic contributions, at intermediate melt fractions the contribution of particle-particle dissipation caused by friction to the matrix rheology is considerable and should be included in regimes where repacking is relevant (i.e., at intermediate melt fractions above the maximum close packing where relative motion between particles isn’t entirely restricted).